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Calculus Examples
Step 1
Write as a function.
Step 2
Step 2.1
Differentiate using the Product Rule which states that is where and .
Step 2.2
Differentiate using the chain rule, which states that is where and .
Step 2.2.1
To apply the Chain Rule, set as .
Step 2.2.2
Differentiate using the Exponential Rule which states that is where =.
Step 2.2.3
Replace all occurrences of with .
Step 2.3
Differentiate.
Step 2.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.3.2
Differentiate using the Power Rule which states that is where .
Step 2.3.3
Simplify the expression.
Step 2.3.3.1
Multiply by .
Step 2.3.3.2
Move to the left of .
Step 2.3.3.3
Rewrite as .
Step 2.3.4
Differentiate using the Power Rule which states that is where .
Step 2.3.5
Multiply by .
Step 2.4
Simplify.
Step 2.4.1
Reorder terms.
Step 2.4.2
Reorder factors in .
Step 3
Step 3.1
By the Sum Rule, the derivative of with respect to is .
Step 3.2
Evaluate .
Step 3.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 3.2.2
Differentiate using the Product Rule which states that is where and .
Step 3.2.3
Differentiate using the chain rule, which states that is where and .
Step 3.2.3.1
To apply the Chain Rule, set as .
Step 3.2.3.2
Differentiate using the Exponential Rule which states that is where =.
Step 3.2.3.3
Replace all occurrences of with .
Step 3.2.4
Since is constant with respect to , the derivative of with respect to is .
Step 3.2.5
Differentiate using the Power Rule which states that is where .
Step 3.2.6
Differentiate using the Power Rule which states that is where .
Step 3.2.7
Multiply by .
Step 3.2.8
Move to the left of .
Step 3.2.9
Rewrite as .
Step 3.2.10
Multiply by .
Step 3.3
Evaluate .
Step 3.3.1
Differentiate using the chain rule, which states that is where and .
Step 3.3.1.1
To apply the Chain Rule, set as .
Step 3.3.1.2
Differentiate using the Exponential Rule which states that is where =.
Step 3.3.1.3
Replace all occurrences of with .
Step 3.3.2
Since is constant with respect to , the derivative of with respect to is .
Step 3.3.3
Differentiate using the Power Rule which states that is where .
Step 3.3.4
Multiply by .
Step 3.3.5
Move to the left of .
Step 3.3.6
Rewrite as .
Step 3.4
Simplify.
Step 3.4.1
Apply the distributive property.
Step 3.4.2
Combine terms.
Step 3.4.2.1
Multiply by .
Step 3.4.2.2
Multiply by .
Step 3.4.2.3
Subtract from .
Step 3.4.3
Reorder terms.
Step 3.4.4
Reorder factors in .
Step 4
To find the local maximum and minimum values of the function, set the derivative equal to and solve.
Step 5
Step 5.1
Find the first derivative.
Step 5.1.1
Differentiate using the Product Rule which states that is where and .
Step 5.1.2
Differentiate using the chain rule, which states that is where and .
Step 5.1.2.1
To apply the Chain Rule, set as .
Step 5.1.2.2
Differentiate using the Exponential Rule which states that is where =.
Step 5.1.2.3
Replace all occurrences of with .
Step 5.1.3
Differentiate.
Step 5.1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 5.1.3.2
Differentiate using the Power Rule which states that is where .
Step 5.1.3.3
Simplify the expression.
Step 5.1.3.3.1
Multiply by .
Step 5.1.3.3.2
Move to the left of .
Step 5.1.3.3.3
Rewrite as .
Step 5.1.3.4
Differentiate using the Power Rule which states that is where .
Step 5.1.3.5
Multiply by .
Step 5.1.4
Simplify.
Step 5.1.4.1
Reorder terms.
Step 5.1.4.2
Reorder factors in .
Step 5.2
The first derivative of with respect to is .
Step 6
Step 6.1
Set the first derivative equal to .
Step 6.2
Factor out of .
Step 6.2.1
Factor out of .
Step 6.2.2
Multiply by .
Step 6.2.3
Factor out of .
Step 6.3
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 6.4
Set equal to and solve for .
Step 6.4.1
Set equal to .
Step 6.4.2
Solve for .
Step 6.4.2.1
Take the natural logarithm of both sides of the equation to remove the variable from the exponent.
Step 6.4.2.2
The equation cannot be solved because is undefined.
Undefined
Step 6.4.2.3
There is no solution for
No solution
No solution
No solution
Step 6.5
Set equal to and solve for .
Step 6.5.1
Set equal to .
Step 6.5.2
Solve for .
Step 6.5.2.1
Subtract from both sides of the equation.
Step 6.5.2.2
Divide each term in by and simplify.
Step 6.5.2.2.1
Divide each term in by .
Step 6.5.2.2.2
Simplify the left side.
Step 6.5.2.2.2.1
Dividing two negative values results in a positive value.
Step 6.5.2.2.2.2
Divide by .
Step 6.5.2.2.3
Simplify the right side.
Step 6.5.2.2.3.1
Divide by .
Step 6.6
The final solution is all the values that make true.
Step 7
Step 7.1
The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.
Step 8
Critical points to evaluate.
Step 9
Evaluate the second derivative at . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.
Step 10
Step 10.1
Simplify each term.
Step 10.1.1
Multiply by .
Step 10.1.2
Multiply by .
Step 10.1.3
Rewrite the expression using the negative exponent rule .
Step 10.1.4
Multiply by .
Step 10.1.5
Rewrite the expression using the negative exponent rule .
Step 10.1.6
Combine and .
Step 10.1.7
Move the negative in front of the fraction.
Step 10.2
Combine fractions.
Step 10.2.1
Combine the numerators over the common denominator.
Step 10.2.2
Simplify the expression.
Step 10.2.2.1
Subtract from .
Step 10.2.2.2
Move the negative in front of the fraction.
Step 11
is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.
is a local maximum
Step 12
Step 12.1
Replace the variable with in the expression.
Step 12.2
Simplify the result.
Step 12.2.1
Multiply by .
Step 12.2.2
Multiply by .
Step 12.2.3
Rewrite the expression using the negative exponent rule .
Step 12.2.4
The final answer is .
Step 13
These are the local extrema for .
is a local maxima
Step 14